《信号与系统 Signals and Systems》课程教学资料（英文版）lecture 13 SAMPLING

Signals and systems Fall 2003 Lecture #13 21 October 2003 The Concept and representation of periodic Sampling of a ct signal Analysis of Sampling in the Frequency Domain 3. The Sampling Theorem -the Nyquist Rate 4. In the Time Domain Interpolation Undersampling and Aliasing

1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling in the Frequency Domain 3. The Sampling Theorem — the Nyquist Rate 4. In the Time Domain: Interpolation 5. Undersampling and Aliasing Signals and Systems Fall 2003 Lecture #13 21 October 2003

SAMPLING We live in a continuous-time world most of the signals we encounter are CT signals, e.g. x(t). How do we convert them into DT signals x[n? Sampling, taking snap shots of x(t)every Second T-sampling period xn=x(nn,n=.-1,0, 1, 2,.-regularly spaced samples Applications and examples Digital Processing of signals Strobe mages in Newspapers Sampling oscilloscope How do we perform sampling?

We live in a continuous-time world: most of the signals we encounter are CT signals, e.g. x ( t). How do we convert them into DT signals x[n]? SAMPLING How do we perform samplin g ? — Sampling, taking snap shots of x ( t) every T seconds. T – sampling period x[n] ≡ x (nT), n = ..., -1, 0, 1, 2, ... — regularly spaced samples Applications and Examples — Digital Processing of Signals — Strobe — Images in Newspapers — Sampling Oscilloscope — …

Why/when Would a Set of samples Be adequate? Observation: Lots of signals have the same samples X3()x1()x2( -3T -2T by sampling we throw out lots of information all values of x(t) between sampling points are lost Key Question for Sampling Under what conditions can we reconstruct the original CT signal x(t) from its samples

Why/When Would a Set of Samples Be Adequate? • Observation: Lots of signals have the same samples • By sampling we throw out lots of information – all values of x ( t) between sampling points are lost. • Key Question for Sampling: Under what conditions can we reconstruct the original CT signal x ( t) from its samples?

Impulse Sampling- Multiplying x(t) by the sampling function p(t)=∑6(t-n) xp()=(1)=∑(106t-nn)=∑a(nm)6(t-n) n=-∞ n=-0o p() X(tH Xp(t) X() p() T X(0) Xp(t)

Impulse Sampling — Multiplying x(t) by the sampling function

Analysis of sampling in the Frequency domain Multiplication Property Xp(jw)=oX(w)* plow) 2丌 P)=r∑6a-ka) k 2丌 T= Sampling Frequency Important to note:ocl/T )=7∑x)*6(-ka) k=-∞ ∑X(a-k)

Analysis of Sampling in the Frequency Domain Import ant to note: ωs∝1/ T

lustration of sampling in the frequency-domain for a band-limited (X(o=0 for a> Om) signal XGo) Pgo Plu) drawn assuming-2a 2 30s0 s-Wm>Wm gjo)= X(o) *P(o)/2T 0. 's> 2WM 人入入人入入 No overlap between shifted spectra "OM 0 COM1 Os Os-CM)

Illustration of sampling in the frequency-domain for a band-limited (X(j ω)=0 for | ω|> ωM) signal No overlap bet ween shifted spectra

Reconstruction of x(t) from sampled signals p()=∑6t-n) n=-0 x(0)H(o) x(t) 0 OM Xp (jo) 0s>20M OM OM<0c<(0)s If there is no overlap between Oc shifted spectra, a LPF can Xr gjo) reproduce x(t) from x(t)

Reconstruction of x(t) from sampled signals If there is no overlap between shifted spectra, a LPF can reproduce x(t) from xp(t)

The sampling Theorem Suppose x(t) is bandlimited, so that Xlw)=0 for W>wM Then x(t)is uniquely determined by its samples x(nn)) if Ws> 2wM= The Nyquist rate where ws

The Sampling Theorem Suppose x ( t) is bandlimited, so that Then x ( t) is uniquely determined by its samples { x (nT)} if

Observations on Sampling don' t sample with impulses x()-+(a/ ho(0 (1)In practice, we obviously Xo(t) or implement ideal lowpass 0 filt One practical example x() The Zero-Order hold Xo(t=Xp(t)=*ho(t)

Observations on Sampling (1) In practice, we obviously don’t sample with impulses or implement ideal lowpass filters. — One practical example: The Zero-Order Hold

Observations( Continued (2)Sampling is fundamentally a time-varying operation, since we multiply x(t with a time-varying function p(t). However, p()=∑8-nT) Hojo) Xr(t) oM 20M) ()What ifos< 2OM? Something different: more later

Observations (Continued) (2) Sampling is fundamentally a time-varying operation, since we multiply x ( t) with a time-varying function p ( t). However, is the identity system (which is TI) for bandlimited x ( t) satisfying the sampling theorem ( ωs > 2 ω M). (3) What if ωs ≤ 2 ω M? Something different: more later